Hidden parameters in quantum mechanics. Hidden parameters in quantum mechanics and Bell's theorem. How gamers choose numbers

HIDDEN PARAMETERS- hypothetical additional variables that are currently unknown, the values ​​of which should completely characterize the state of the system and determine its future more fully than quantum mechanics. state vector. It is believed that with the help of S. p. from statistical. Descriptions of micro-objects can be moved to dynamic ones. patterns, in which the physical bodies themselves are uniquely connected in time. values, and not their statistics. distributions (see Causality). WITH. n. are usually considered diff. fields or coordinates and momenta of smaller, constituent parts of quantum particles. However, after the discovery (of the composite particles of hadrons), it turned out that their behavior obeys, like the behavior of hadrons themselves.

According to von Neumann’s theorem, not a single theory with quantum mechanics can reproduce all the consequences of quantum mechanics, however, as it later turned out, J. von Neumann’s proof was based on assumptions that, generally speaking, are not necessary for any model S.p. A weighty argument in favor of the existence of S. p. was put forward by A. Einstein, B. Podolsky, and N. Rosen in 1935 (the so-called Einstein - Podolsky - Rosen paradox), the essence of which is that certain characteristics of quantum particles (in particular, spin projections) can be measured without subjecting the particles to force. A new incentive to experiment. testing of the Einstein-Podolsky-Rosen paradox was proven in 1951 Bell's inequalities, which made it possible to conduct direct experiments. tests of the hypothesis about the S.P. These inequalities demonstrate the difference between the predictions of quantum mechanics and any theories of S.P. that do not allow the existence of physical phenomena. processes propagating at superluminal speeds. Experiments carried out in a number of laboratories around the world have confirmed the predictions of quantum mechanics about the existence of stronger correlations between particles than predicted by any local theories of quantum mechanics. According to these theories, the results of an experiment conducted on one of the particles are determined only by this experiment itself and do not depend on the results experiment, which can be carried out on another particle that is not associated with the first by force interactions.

Lit.: 1) Sudbury A., Quantum mechanics and elementary particles, trans. from English, M., 1989; 2) A. A. Grib, Bell’s inequalities and experimental verification of quantum correlations at macroscopic distances, UFN, 1984, v. 142, p. 619; 3) Spassky B.I., Moskovsky A.V., On nonlocality in quantum physics, UFN, 1984, vol. 142, p. 599; 4) Bom D., On the possibility of interpreting quantum mechanics based on ideas about “hidden” parameters, in the collection: Questions of causality in quantum mechanics, M., 1955, p. 34. G. Ya. Myakishev.

The principle of sufficient reason is key to the program of expanding physics to the scale of the universe: it seeks to provide a rational explanation for every choice that nature makes. The free, causeless behavior of quantum systems contradicts this principle.

Can it be observed in quantum physics? It depends on whether quantum mechanics can be extended to the entire universe and offer the most fundamental description of nature possible—or whether quantum mechanics is merely an approximation to another cosmological theory. If we can extend quantum theory to the universe, the free will theorem becomes applicable on cosmological scales. Since we assume that there is no theory more fundamental than quantum, we imply that nature is truly free. The freedom of quantum systems on cosmological scales would be a limitation of the principle of sufficient reason, because there cannot be a rational or sufficient reason for the many cases of free behavior of quantum systems.

But in proposing an extension of quantum mechanics, we make a cosmological mistake: we apply the theory beyond the boundaries of the region in which it can be tested. A more cautious step would be to consider the hypothesis that quantum physics is an approximation valid only for small subsystems. To determine whether a quantum system is present elsewhere in the Universe or whether a quantum description can be applied to a theory of the entire Universe, additional information is needed.

Can there be a deterministic cosmological theory that reduces to quantum physics when we isolate a subsystem and neglect everything else in the world? Yes. But this comes at a high price. According to this theory, probability in quantum theory arises only due to the neglect of the influence of the entire Universe. Probabilities will give way to certain predictions at the level of the Universe. In cosmological theory, quantum uncertainties appear when trying to describe a small part of the Universe.

The theory is called the theory of hidden parameters, since quantum uncertainties are eliminated by information about the Universe that is hidden from the experimenter working with a closed quantum system. Theories of this kind serve to make predictions for quantum phenomena that are consistent with the predictions of traditional quantum physics. So, a similar solution to the problem of quantum mechanics is possible. Moreover, if determinism is restored by extending quantum theory to the entire Universe, the hidden parameters are associated not with a refined description of the individual elements of the quantum system, but with the interaction of the system with the rest of the Universe. We can call them hidden relational parameters. According to the principle of maximum freedom, described in the previous chapter, quantum theory is probabilistic and its internal uncertainties are maximum. In other words, the information about the state of the atom that we need to restore determinism, and which is encoded in the relationship of this atom with the entire Universe, is maximum. That is, the properties of each particle are maximally encoded using hidden connections with the Universe as a whole. The task of clarifying the meaning of quantum theory in the search for a new cosmological theory is key.

What is the price of the “entrance ticket”? Abandonment of the principle of the relativity of simultaneity and a return to a picture of the world in which the absolute definition of simultaneity is valid throughout the entire Universe.

We must proceed with caution because we do not want to conflict with the theory of relativity, which has had many successful applications. Among them, quantum field theory is a successful unification special theory relativity (STR) and quantum theory. It is this that underlies the standard model of particle physics and allows us to obtain many accurate predictions that are confirmed by experiments.

But quantum field theory is not without problems. Among them is the complex manipulation of infinite quantities that must be done before a prediction can be obtained. Moreover, quantum field theory inherits all the conceptual problems of quantum theory and does not offer anything new to solve them. Old problems, together with new problems of infinity, show that quantum field theory is an approximation to a deeper theory.

Many physicists, starting with Einstein, dreamed of going beyond quantum field theory and finding a theory that would give Full description each experiment (which, as we have seen, is impossible within the framework of quantum theory). This led to irremovable contradiction between quantum mechanics and SRT. Before we move on to bringing time back into physics, we need to understand what this contradiction is.

There is an opinion that the inability of quantum theory to present a picture of what is happening in a particular experiment is one of its advantages, and not at all a defect. Niels Bohr argued (see Chapter 7) that the goal of physics is to create a language in which we can communicate to each other how we experimented with atomic systems and what results we obtained.

I find this unconvincing. By the way, I have the same feelings in relation to some modern theorists who convince me that quantum mechanics deals not with the physical world, but with information about it. They argue that quantum states do not correspond to physical reality, but simply encode information about the system that we as observers can obtain. This smart people, and I like to argue with them, but I'm afraid they underestimate the science. If quantum mechanics is just an algorithm for predicting probabilities, can we come up with anything better? In the end, something happens in a specific experiment, and only this is the reality called an electron or a photon. Are we able to describe the existence of individual electrons in mathematical language? There is perhaps no principle guaranteeing that the reality of every subatomic process must be intelligible to man and can be formulated in human language or mathematics. But shouldn't we try? I'm on Einstein's side here. I believe that there is an objective physical reality and something describable happens when an electron jumps from one energy level to another. I will try to construct a theory capable of giving such a description.

The theory of hidden parameters was first presented by Duke Louis de Broglie at the famous V Solvay Congress in 1927, shortly after quantum mechanics acquired its final formulation. De Broglie was inspired by Einstein's idea of ​​the duality of wave and particle properties (see Chapter 7). De Broglie's theory solved the wave-particle puzzle in the simplest way. He argued that both a particle and a wave physically exist. Previously, in a 1924 dissertation, he had written that wave-particle duality was universal, so that particles such as electrons were also waves. In 1927, de Broglie stated that these waves propagate as on the surface of water, interfering with each other. The particle corresponds to a wave. In addition to electrostatic, magnetic and gravitational forces, quantum force acts on particles. It attracts particles to the wave crest. Therefore, on average, the particles will most likely be located there, but this relationship is probabilistic. Why? Because we don't know where the particle was at first. And if so, we cannot predict where she will end up next. The hidden variable in this case is the exact position of the particle.

Later, John Bell proposed to call de Broglie's theory the theory of real variables (beables), in contrast to the quantum theory of observable variables. Real variables are always present, in contrast to observed ones: the latter arise as a result of experiment. According to de Broglie, both particles and waves are real. A particle always occupies a certain position in space, even if quantum theory cannot accurately predict it.

De Broglie's theory, in which both particles and waves are real, was not widely accepted. In 1932, the great mathematician John von Neumann published a book in which he argued that the existence of hidden parameters is impossible. A few years later, Greta Hermann, a young German mathematician, pointed out the vulnerability of von Neumann's proof. Apparently, he made a mistake, initially believing that what he wanted to prove was proven (that is, he passed off the assumption as an axiom and deceived himself and others). But Herman’s work was ignored.

Two decades passed before the error was discovered again. In the early 50s American physicist David Bohm wrote a textbook on quantum mechanics. Bohm, independently of de Broglie, discovered the theory of hidden parameters, but when he sent the article to the editor of the journal, he was refused: his calculations contradicted von Neumann's well-known proof of the impossibility of hidden parameters. Bohm quickly found the error in von Neumann. Since then, few people have used the de Broglie–Bohm approach to quantum mechanics in their work. This is one of the views on the foundations of quantum theory that is still discussed today.

Thanks to the de Broglie-Bohm theory, we understand that hidden parameter theories represent a solution to the paradoxes of quantum theory. Many features of this theory turned out to be inherent in any theory of hidden parameters.

The de Broglie–Bohm theory has an ambivalent relationship to the theory of relativity. Its statistical predictions are consistent with quantum mechanics and do not contradict special relativity (for example, the principle of the relativity of simultaneity). But unlike quantum mechanics, de Broglie-Bohm theory offers more than statistical predictions: it provides a detailed physical picture of what happens in each experiment. A wave that changes in time affects the movement of particles and violates the relativity of simultaneity: the law according to which a wave affects the movement of a particle can only be true in one of the reference frames associated with the observer. Thus, if we accept the de Broglie–Bohm hidden parameter theory as an explanation of quantum phenomena, we must take it on faith that there is a dedicated observer whose clock shows a dedicated physical time.

This attitude towards the theory of relativity extends to any theory of hidden parameters. Statistical predictions that are consistent with quantum mechanics are also consistent with the theory of relativity. But any detailed picture of phenomena violates the principle of relativity and will have an interpretation in a system with only one observer.

The de Broglie-Bohm theory is not suitable for the role of cosmology: it does not meet our criteria, namely the requirement that actions be mutual for both parties. The wave affects the particles, but the particle has no effect on the wave. However, there is also alternative theory hidden settings, which resolves this issue.

Convinced, like Einstein, that there was another, deeper theory underlying quantum theory, I have been inventing theories of hidden parameters since my days as a student. Every few years I would put all my work aside and try to solve this the most important problem. For many years I have been developing an approach based on the theory of hidden parameters proposed by the Princeton mathematician Edward Nelson. This approach worked, but it had an element of artificiality: to reproduce the predictions of quantum mechanics, certain forces had to be precisely balanced. In 2006, I wrote an article explaining the unnaturalness of the theory due to technical reasons, and abandoned this approach.

One evening (it was early autumn 2010) I walked into a cafe, opened my notebook and thought about my many unsuccessful attempts to go beyond quantum mechanics. And I remembered the statistical interpretation of quantum mechanics. Instead of trying to describe what happens in a particular experiment, it describes an imaginary collection of everything that should happen. Einstein put it this way: “The attempt to present a quantum theoretical description as a complete description of individual systems leads to unnatural theoretical interpretations, which become unnecessary if one accepts that the description refers to ensembles (or collections) of systems, and not to individual systems.”

Consider a lone electron orbiting a proton in a hydrogen atom. According to the authors of the statistical interpretation, the wave is associated not with a single atom, but with an imaginary collection of copies of the atom. Different samples in the collection have electrons different position in space. And if you observe a hydrogen atom, the result will be the same as if you randomly selected an atom from an imaginary collection. The wave gives the probability of finding the electron in all the different positions.

I liked this idea for a long time, but now it seemed crazy. How can an imaginary set of atoms influence measurements on one real atom? This would contradict the principle that nothing outside the Universe can influence what is inside it. And I asked myself: can I replace the imaginary set with a collection of real atoms? To be real, they must exist somewhere. There are a great many hydrogen atoms in the Universe. Can they form the “collection” that the static interpretation of quantum mechanics treats?

Imagine that all the hydrogen atoms in the universe are playing a game. Each atom recognizes that the others are in a similar situation and have a similar history. By “similar” I mean that they will be described probabilistically, using the same quantum state. Two particles in the quantum world can have the same history and be described by the same quantum state, but differ in the exact values ​​of real variables, such as their position. When two atoms have a similar history, one copies the properties of the other, including the exact values ​​of real variables. Atoms do not have to be close to each other to copy properties.

This is a non-local game, but any theory of hidden parameters must express the fact that the laws of quantum physics are non-local. Although the idea may seem crazy, it is less crazy than the idea of ​​an imaginary collection of atoms influencing atoms in real world. I undertook to develop this idea.

One of the copied properties is the position of the electron relative to the proton. Therefore, the position of an electron in a particular atom will change as it copies the position of electrons in other atoms in the universe. As a result of these jumps, measuring the position of an electron in a particular atom will be equivalent to choosing an atom at random from a collection of all similar atoms, replacing the quantum state. To make this work, I came up with copying rules that lead to predictions for the atom that agree exactly with those of quantum mechanics.

And then I realized something that made me immensely happy. What if the system has no analogues in the Universe? The copying cannot continue and the results of quantum mechanics will not be reproduced. This would explain why quantum mechanics does not apply to complex systems like us humans or cats: we are unique. This resolved long-standing paradoxes that arise when quantum mechanics is applied to large objects such as cats and observers. The strange properties of quantum systems are limited to atomic systems because the latter are found in great abundance throughout the Universe. Quantum uncertainties arise because these systems constantly copy each other's properties.

I call this the real statistical interpretation of quantum mechanics (or the “white squirrel interpretation,” after the albino squirrels that are occasionally found in Toronto parks). Imagine that all gray squirrels are sufficiently similar to each other that quantum mechanics applies to them. Find one gray squirrel and you'll likely find more soon. But the flashing white squirrel does not seem to have a single copy, and therefore it is not a quantum mechanical squirrel. She (like me or you) can be considered as having unique properties and having no analogues in the Universe.

Playing with jumping electrons violates the principles of special relativity. Instantaneous jumps over arbitrarily large distances require the concept of simultaneous events separated by large distances. This, in turn, implies the transfer of information at speeds exceeding the speed of light. However, statistical predictions are consistent with quantum theory and can be reconciled with relativity. And yet in this picture there is a highlighted simultaneity - and, therefore, a highlighted time scale, as in the de Broglie-Bohm theory.

Both of the latent parameter theories described above adhere to the principle of sufficient reason. There is a detailed picture of what happens in individual events, and it explains what is considered uncertain in quantum mechanics. But the price for this is a violation of the principles of the theory of relativity. This is a high price.

Could there be a hidden parameter theory that is compatible with the principles of relativity? No. It would violate the free will theorem, which implies that as long as its conditions are met, it is impossible to determine what will happen to a quantum system (and therefore that no hidden parameters exist). One of these conditions is the relativity of simultaneity. Bell's theorem also excludes local hidden parameters (local in the sense that they are causally connected and exchange information at a transmission speed less than the speed of light). But the theory of hidden parameters is possible if it violates the principle of relativity.

As long as we are only testing the predictions of quantum mechanics on a statistical level, there is no need to wonder what the correlations actually are. But if we try to describe the transfer of information within each entangled pair, the concept of instantaneous communication is required. And if we try to go beyond the statistical predictions of quantum theory and move to the theory of hidden parameters, we will come into conflict with the principle of the relativity of simultaneity.

To describe correlations, latent parameter theory must adopt a definition of simultaneity from the point of view of a single dedicated observer. This, in turn, means that there is a distinguished concept of position of rest and, therefore, that motion is absolute. It takes on absolute meaning because you can state who is moving relative to whom (let's call this character Aristotle). Aristotle is in a state of rest, and all that he sees as a moving body is a real moving body. That's the whole conversation.

In other words, Einstein was wrong. And Newton. And Galileo. There is no relativity in movement.

This is our choice. Either quantum mechanics is a definitive theory and there is no way to penetrate its statistical veil to reach a deeper level of description of nature, or Aristotle was right and distinguished systems of motion and rest exist.

See: Bacciagaluppi, Guido, and Antony Valentini Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference. New York: Cambridge University Press, 2009.

See: Bell, John S. Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy. New York: Cambridge University Press, 2004.

Neumann, John von Mathematische Grundlagen der Quantenmechanik. Berlin, Julius Springer Verlag, 1932, pp. 167 ff.; Neumann, John von Mathematical Foundations of Quantum Mechanics. Princeton, NJ: Princeton University Press, 1996.

Hermann, Grete Die Naturphilosophischen Grundlagen der Quantenmechanik // Abhandlungen der Fries’schen Schule (1935).

Bohm, David Quantum Theory. New York: Prentice Hall, 1951.

Bohm, David A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. II // Phys. Rev. 85:2, 180–193 (1952).

Valentini, Antony Hidden Variables and the Large-scale Structures of Space=Time / In: Einstein, Relativity and Absolute Simultaneity. Eds. Craig, W. L., and Q. Smith. London: Routledge, 2008. Pp. 125–155.

Smolin, Lee Could Quantum Mechanics Be an Approximation to Another Theory? // arXiv: quant-ph/0609109v1 (2006).

Einstein, Albert Remarks to the Essays Appearing in This Collective Volume / In: Albert Einstein: Philosopher-Scientist. Ed. P. A. Schilpp. New York: Tudor, 1951. P. 671.

See: Smolin, Lee A Real Ensemble Interpretation of Quantum Mechanics // arXiv:1104.2822v1 (2011).

Alexey Paevsky

First, it’s worth debunking one myth. Einstein never said the words “God does not play dice.” In fact, he wrote to Max Born regarding Heisenberg's uncertainty principle: “Quantum mechanics is truly impressive. But an inner voice tells me that this is not ideal yet. This theory says a lot, but still does not bring us closer to unraveling the secret of the Almighty. At least I'm sure He doesn't roll the dice."

However, he also wrote to Bohr: “You believe in a God who plays dice, and I believe in complete regularity in the world of objectively existing things.” That is, in this sense, Einstein spoke about determinism, that at any moment you can calculate the position of any particle in the Universe. As Heisenberg showed us, this is not so.

But nevertheless, this element is very important. Indeed, paradoxically, the greatest physicist of the 20th century, Albert Einstein, who broke the physics of the past with his articles at the beginning of the century, later turned out to be a zealous rival of the even newer one, quantum mechanics. All of him scientific intuition protested against describing the phenomena of the microworld in terms of probability theory and wave functions. But it’s hard to go against the facts - and it turned out that any measurement of a system of quantum objects changes it.

Einstein tried to “get out” and suggested that there are some hidden parameters in quantum mechanics. For example, there are certain sub-instruments that can be used to measure the state of a quantum object and not change it. As a result of such reflections, in 1935, together with Boris Podolsky and Nathan Rosen, Einstein formulated the principle of locality.

Albert Einstein

This principle states that the results of any experiment can only be influenced by objects located close to the place where it is carried out. Moreover, the movement of all particles can be described without involving the methods of probability theory and wave functions, introducing into the theory those very “hidden parameters” that cannot be measured using conventional instruments.

Bell's theory

John Bell

Almost 30 years have passed, and John Bell has theoretically shown that it is actually possible to conduct an experiment whose results will determine whether quantum mechanical objects are truly described by probability distribution wave functions as they are, or whether there is a hidden parameter that allows them to be accurately described position and momentum, like a billiard ball in Newton's theory.

At that time, there were no technical means to conduct such an experiment: first, one had to learn how to obtain quantum entangled pairs of particles. These are particles that are in a single quantum state, and if they are separated to any distance, they still instantly sense what is happening to each other. We wrote a little about the practical use of the entanglement effect in quantum teleportation.

In addition, it is necessary to quickly and accurately measure the states of these particles. Everything is fine here too, we can do that.

However, there is a third condition in order to test Bell's theory: you need to collect large statistics on random changes in the settings of the experimental setup. That is, it was necessary to conduct a large number of experiments, the parameters of which would be set completely randomly.

And here there is a problem: all our random number generators use quantum methods - and here we can introduce those same hidden parameters into the experiment ourselves.

How gamers choose numbers

And here the researchers were saved by the principle described in the joke:

“One programmer approaches another and says:

– Vasya, I need a random number generator.

“One hundred and sixty-four!”

The generation of random numbers was entrusted to gamers. True, a person does not actually choose numbers at random, but this is precisely what the researchers played on.

They created a browser game in which the player's task was to obtain the longest sequence of zeros and ones as possible - while through his actions the player trained a neural network that tried to guess which number the person would choose.

This greatly increased the “purity” of randomness, and if you take into account the breadth of coverage of the game in the press and reposts on social networks, then up to hundreds of thousands of people were playing the game at the same time, the flow of numbers reached a thousand bits per second, and more than a hundred million random choices had already been created.

This truly random data, which was used in 13 experimental setups in which different quantum objects were entangled (one with qubits, two with atoms, ten with photons), was enough to show that Einstein was wrong after all. .

There are no hidden parameters in quantum mechanics. Statistics have shown this. This means that the quantum world remains truly quantum.

"God doesn't play dice with the universe."

With these words, Albert Einstein challenged his colleagues who were developing a new theory - quantum mechanics. In his opinion, Heisenberg's uncertainty principle and Schrödinger's equation introduced unhealthy uncertainty into the microworld. He was sure that the Creator could not allow the world of electrons to be so strikingly different from the familiar world of Newtonian billiard balls. In fact, throughout for long years Einstein played devil's advocate with regard to quantum mechanics, inventing clever paradoxes designed to lead the creators of the new theory into a dead end. In doing so, however, he did a good job of seriously perplexing the theoreticians of the opposing camp with his paradoxes and forcing them to think deeply about how to resolve them, which is always useful when a new field of knowledge is being developed.

There is a strange irony of fate in the fact that Einstein went down in history as a principled opponent of quantum mechanics, although initially he himself stood at its origins. In particular, Nobel Prize in physics for 1921, he received it not for the theory of relativity at all, but for the explanation of the photoelectric effect on the basis of new quantum concepts that literally swept the scientific world at the beginning of the twentieth century.

Most of all, Einstein protested against the need to describe the phenomena of the microworld in terms of probabilities and wave functions ( cm. Quantum mechanics), and not from the usual position of coordinates and particle velocities. That's what he meant by "rolling the dice." He recognized that describing the movement of electrons in terms of their speeds and coordinates contradicts the uncertainty principle. But, Einstein argued, there must be some other variables or parameters, taking into account which the quantum mechanical picture of the microworld will return to the path of integrity and determinism. That is, he insisted, it only seems to us that God is playing dice with us, because we do not understand everything. Thus, he was the first to formulate hypotheses for the latent variable in the equations of quantum mechanics. It consists in the fact that in fact electrons have fixed coordinates and speed, like Newton’s billiard balls, and the uncertainty principle and the probabilistic approach to their determination within the framework of quantum mechanics are the result of the incompleteness of the theory itself, which is why it does not allow them determine for sure.

The hidden variable theory can be visualized something like this: the physical justification for the uncertainty principle is that the characteristics of a quantum object, such as an electron, can only be measured through its interaction with another quantum object; in this case, the state of the measured object will change. But perhaps there is some other way of measuring using tools that are still unknown to us. These instruments (let's call them "subelectrons") will probably interact with quantum objects without changing their properties, and the uncertainty principle will not apply to such measurements. Although there was no actual evidence in favor of hypotheses of this kind, they loomed ghostly on the sidelines of the main path of development of quantum mechanics - mainly, I believe, due to the psychological discomfort experienced by many scientists due to the need to abandon established Newtonian ideas about the structure of the Universe.

And in 1964, John Bell obtained a new theoretical result that was unexpected for many. He proved that it was possible to conduct a specific experiment (details in a moment), the results of which would allow us to determine whether quantum mechanical objects are truly described by probability distribution wave functions as they are, or whether there is a hidden parameter that allows their position and momentum to be accurately described as at a Newtonian ball. Bell's theorem, as it is now called, shows that just as there is a hidden parameter in quantum mechanical theory that affects any physical characteristics of a quantum particle, and in the absence of one, it is possible to conduct a serial experiment, the statistical results of which will confirm or refute the presence of hidden parameters in quantum mechanical theory. Relatively speaking, in one case the statistical ratio will be no more than 2:3, and in the other - no less than 3:4.

(Here I want to note parenthetically that in the year that Bell proved his theory to him, I was an undergraduate student at Stanford. Red-bearded, with a thick Irish accent, Bell was hard to miss. I remember standing in the corridor of the science building of the Stanford Line accelerator, and then he came out of his office in a state of extreme excitement and publicly announced that he had just discovered a really important and interesting thing. And although I have no evidence on this score, I would really like to hope that I that day I became an involuntary witness to its discovery.)

However, the experience proposed by Bell turned out to be simple only on paper and at first seemed almost impossible. The experiment should have looked like this: under external influence, the atom should have synchronously emitted two particles, for example two photons, and in opposite directions. After this, it was necessary to capture these particles and instrumentally determine the direction of the spin of each and do this a thousand times in order to accumulate sufficient statistics to confirm or refute the existence of a hidden parameter according to Bell’s theorem (in the language of mathematical statistics, it was necessary to calculate correlation coefficients).

The most unpleasant surprise for everyone after the publication of Bell’s theorem was precisely the need to carry out a colossal series of experiments, which at that time seemed almost impossible, to obtain a statistically reliable picture. However, less than a decade had passed before experimental scientists not only developed and built necessary equipment, but also accumulated a sufficient amount of data for statistical processing. Without going into technical details, I will only say that at that time, in the mid-sixties, the complexity of this task seemed so monstrous that the likelihood of its implementation seemed equal to that of someone planning to put the proverbial million trained monkeys at typewriters in the hope of finding among the fruits of their collective labor is a creation equal to Shakespeare.

When the results of the experiments were summarized in the early 1970s, everything became crystal clear. The wave probability distribution function completely accurately describes the movement of particles from the source to the sensor. Therefore, the equations of wave quantum mechanics do not contain hidden variables. This is the only known case in the history of science when a brilliant theorist proved opportunity experimental testing of hypotheses and gave justification method Such verification, brilliant experimenters, with titanic efforts, carried out a complex, expensive and lengthy experiment, which in the end only confirmed the already dominant theory and did not even introduce anything new into it, as a result of which everyone felt cruelly deceived in their expectations!

However, not all the work was in vain. More recently, scientists and engineers, much to their own surprise, have found a very worthy practical application of Bell’s theorem. The two particles emitted by the source at the Bell facility are coherent(have the same wave phase) because they are emitted synchronously. And this property is now going to be used in cryptography to encrypt highly secret messages sent through two separate channels. When intercepting and trying to decipher a message through one of the channels, coherence is instantly broken (again due to the uncertainty principle), and the message inevitably and instantly self-destructs at the moment the connection between particles is broken.

But Einstein, it seems, was wrong: God is still playing dice with the Universe. Perhaps Einstein should have heeded the advice of his old friend and colleague Niels Bohr, who, in Once again Having heard the old refrain about “playing dice,” he exclaimed: “Albert, finally stop telling God what to do!”

In quantum mechanics

Hidden parameter theory (HPT) is a traditional, but not the only basis for constructing various types Bell's theorem. A starting point may also be to recognize the existence of a positive definite probability distribution function. Based on this assumption, without resorting to additional assumptions, the work formulates and proves Bell's paradoxes various types. On specific example it is shown that formal quantum calculation sometimes gives negative values appearing in the proof of joint probabilities. An attempt is made to clarify the physical meaning of this result and an algorithm for measuring negative joint probabilities of this type is proposed.

Since the laws of quantum theory predict the results of an experiment, generally speaking, only statistically, then, based on the classical point of view, one could assume that there are hidden parameters which, being unobservable in any ordinary experiment, actually determine the result of the experiment, as this has always been previously considered in accordance with the principle of causality. Therefore, an attempt was made to invent such parameters within the framework of quantum mechanics.

In a narrow sense, applicable in quantum mechanics and theoretical physics of the microworld, where the determinism of the laws of macroscopic physics ceases to apply, the theory of hidden parameters has served as an important tool of knowledge.

But the significance of the approach to the theory of hidden parameters, undertaken within the framework of the study of the microworld and quantum mechanical paradoxes, is not limited to this range of phenomena. A broader, truly philosophical interpretation of the reasons why this phenomenon occurs in our world is possible.

In the philosophy of knowledge

However, the raised issue of hidden parameters relates not only to narrow physical problems. It relates to the general methodology of cognition. A short excerpt from a treatise on understanding written by A. M. Nikiforov helps to understand the essence of this phenomenon:

First, let's try to understand what understanding is at the usual everyday level. We can say that understanding is the process of reducing the incomprehensible to the understandable. That is, through accessible logical manipulations, from ideas we understand, we build a representation (model) of something that was previously incomprehensible to us. […] There is another approach to understanding when the existence of a certain entity or substance is declared that has the necessary properties that ensure the existence of the phenomenon of interest to us... It should be noted that this approach underlies the theory of relativity and quantum mechanics, which declare how, but not explain why. […] It must be said that if the first approach is more rigorous and clear, then the second is more powerful, universal and simple... The first approach is widely used in science, and it can be considered dominant, but the second is also used. An example of this is the “hidden parameter theory”[emphasis added], according to which the discrepancy between theory and experiment is removed by introducing a certain hypothetical object. The parameters of this object are substituted into the formula, and it begins to coincide with the experiment.

In quantum mechanics, this theory has a significant range of application, although it is not generally accepted.

Historical example

For many centuries, Euclid's geometry was considered the unshakable rock of science. For a long time Before the start of physical research of the microworld and astrophysical measurements, there was no reason to consider it incomplete. However, the situation changed in the first decade of the 20th century. A conceptual crisis was growing in physics, which Albert Einstein was able to resolve. Along with the resolution of particular problems - coordinating observations with the predictions of theories of that time ("saving the phenomenon") - in his work together with Niels Bohr, Einstein was able to draw an ingenious conclusion regarding the possibility of the influence of masses on the geometry of space and the speed of a moving object - at speeds commensurate with light. - over the course of local time for a given object.

In geometry, this became an epoch-making theoretical and practical discovery for cosmology, although it echoed the theoretical premises postulated by Hermann Minkowski, but occupied a special place in modern cosmology.

The effect of the real influence of gravity on the geometry of space can be considered a “hidden parameter” in the classical theory of Euclid, however, it was revealed in the theory of Einstein. Reasoning from the point of view of the methodology of cognition: in one conceptual (theoretical) system, a certain parameter can be hidden, but in another it can become revealed, in demand and theoretically justified. In the first case, its “non-disclosure” does not at all mean the absence of this parameter in nature as such. It’s just that this parameter was not significant, and therefore was not found, nor was it introduced by any of the scientists into the “fabric” of this theory.

This situation quite clearly reveals the properties of such “hidden parameters”. This is not a denial of the predecessor theory, but a finding of objective limitations for its predictions. In the case considered above, the physical space is indeed Euclidean with high accuracy in the case of insufficiently strong gravitational fields acting within a given space (which is the earth’s field), but more and more ceases to be so with a huge increase in the gravitational potential. The latter, in observable nature, can only manifest itself in extraterrestrial space objects such as black holes and some other “exotic” space objects.

Notes

Links

• I. Z. Tsekhmistro, V. I. Shtanko and others. “CONCEPT OF INTEGRITY” - CHAPTER 3 CONCEPT OF INTEGRITY AND EXPERIMENT: causality and nonlocality in quantum physics (L. E. Pargamanik)

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